Geometry and Quantum Physics: Proceeding of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, January 9–16, 1999

412 Pages·2000·13.616 MB·English

by  Anton Alekseev (auth.), H. Gausterer, L. Pittner, Harald Grosse (eds.)

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H. Gausterer H. Grosse L. Pittner (Eds.)
Geometry
and Quantum Physics
Proceedings of the
38. Internationale Universita¨tswochen
fu¨r Kern- und Teilchenphysik,
Schladming, Austria, January 9-16, 1999
1 3
Editors
H.Gausterer
L.Pittner
Institutfu¨rTheoretischePhysik,Karl-Franzens-Universita¨t
Universita¨tsplatz5,8010Graz,Austria
HaraldGrosse
Institutfu¨rTheoretischePhysik,Universita¨tWien
Boltzmanngasse5,1090Wien,Austria
SupportedbytheO¨sterreichischeBundesministeriumfu¨rWirtschaft,
VerkehrundKunst,Vienna,Austria
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DieDeutscheBibliothek-CIP-Einheitsaufnahme
Geometryandquantumphysics:proceedingsofthe38.Internationale
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Preface
In many areas of theoretical physics, geometrical and functional analytic
methods are used simultaneously. During the course of our studies and edu-
cation, these methods appeared to be extremely di(cid:11)erent. For classical (cid:12)eld
theories, a di(cid:11)erential complex over smooth manifolds, smooth sections of
vector bundles over manifolds, principal (cid:12)bre bundles, connections, curvatu-
res, etc. are introduced. For quantum (cid:12)eld theory, singular operator-valued
distributionsactonFockspaceandleadtoallthewell-knowndi(cid:14)culties.Af-
tertheworkofAlainConnesandthemanyapplicationsofnon-commutative
geometry to various subjects, we became familiar with, for example, the fact
that on any associative superalgebra at least one di(cid:11)erential calculus exists.
Butclearlytherearemanymoreaspects.Weselectedsevensubjectsandlec-
turerstocoversymplecticgeometry,quantumgravity,stringtheory,anintro-
duction to non-commutative geometry, and the application to fundamental
interactions, to the quantum Hall e(cid:11)ect as well as to physics on q-deformed
spaces.
In the (cid:12)rst contribution, Anton Alexseev uses techniques of symplec-
tic geometry to evaluate certain integrals over manifolds that have special
symmetries. For them the stationary phase approximation becomes exact.
Starting from simple examples, he introduces equivariant cohomology and
sketches the proof of localization formulas of Duistermaat - Heckman type.
The Weil model, well-known from the BRST formalism, is introduced, and a
non-commutative generalization is used to prove group-valued localization.
John Baez elucidates the present status of quantum geometry of space-
time. Since the implementation of constraints within the Hamiltonian ap-
proach to Einstein gravity is complicated, he explains in detail the simplify-
ing BF system connected to the Chern - Simons model. We learn about
spin net works and spin foams, become familiar with triangulations of four-
manifolds,andcalculatespectraofquantumtetrahedra.Hislecturenotesnot
only survey the subject, but also give an extensive list of references arranged
according to nine di(cid:11)erent subjects.
Cesar Gomez then gives an introduction to string theory. Starting from
themodeexpansionheexplainsT-duality,asymmetryrelatingclosedstrings
of radius R and 1/R. Next D-brane ideas are represented.
Both quantum gravity and string theory have their own way of treating
space-time. But there is a further approach that goes by the name of non-
commutative geometry. This is the subject of the chapter by Daniel Kastler.
John Madore introduces a di(cid:11)erential complex over an arbitrary associa-
tive algebra. We learn that, depending on the procedure, a regularization
e(cid:11)ect may result. Attempts to add a gravitational (cid:12)eld are also reviewed.
This introduction is useful to follow the lectures of Daniel Kastler. He
spoke about Connes’ approach to the standard model as well as recent ideas
forincludinggravityasanexternal(cid:12)eld.Hepresentstheinterestingideathat
VI Preface
additional dimensions (cid:12)nally lead via the Higgs e(cid:11)ect to masses of particles,
as well as the recent attempts to use SU (2) for q being a third root of unity
q
to \deduce" the gauge group of the standard model.
JuliusWessappliesideasofnon-commutativegeometrytotheq-deformed
Heisenberg algebra. The spectra of position and momentum are discrete.
Phase space gets a lattice-like structure. Higher-dimensional analogues are
investigated too.
Finally,RuediSeilerexplainsgeometricalpropertiesoftransportinquan-
tum Hall systems. The integer e(cid:11)ect is treated in detail.
Although we learn here about many di(cid:11)erent applications of non-com-
mutative geometry, the hope is that a unique picture of a quantum space-
timemay(cid:12)nallyresult.Quantumtheorychangesourideasongeometry,and
further surprises may come along soon.
We have tried to cover many subjects and were lucky to be supported by
so many excellent lecturers. We hope that the school helped young people
seekinganentrytothesesubjects.Theywillhopefullyremembertheexcellent
atmosphere of the Schladming Winter School 1999.
Atthispointwealsowanttoexpressourthankstothemainsponsorofthe
School, the Austrian Ministry for Science and Transportation. In addition,
we grateful acknowledge the generous support by the Government of Styria
and the Town of Schladming. Valuable help towards the organization was
received from the Wirtschaftskammer Steiermark (Sektion Industrie), Steyr-
Daimler-Puch AG, Ricoh-Austria, and Styria Online.
Organizing the 1999 Schladming Winter School, and making it the suc-
cessful event that it was, would not have been possible without the help of
a number of colleagues and graduate students from our institute. Without
naming them all we want to acknowledge the traditionally good coopera-
tion within the organizing committee and beyond, which once again guaran-
teed the smooth running of all organizational, technical, and social matters.
Thanks are due to Wolfgang Schweiger for his valuable technical assistance.
Finally we should like to express our sincere thanks to Miss Sabine Fuchs
for carrying out the secretarial work and for (cid:12)nalizing the text and layout of
these proceedings.
Graz, November 1999 H. Gausterer, H. Grosse, L. Pittner
Contents
Notes on Equivariant Localization
AntonAlekseev . . . . . . . . . . . . . . . . . . . . . . . . 1
An Introduction to Spin Foam Models
of BF Theory and Quantum Gravity
JohnC.Baez . . . . . . . . . . . . . . . . . . . . . . . . . 25
T-Duality and the Gravitational Description
of Gauge Theories
CesarGomez and PedroSilva . . . . . . . . . . . . . . . . . . 95
Noncommutative Geometry and Basic Physics
DanielKastler . . . . . . . . . . . . . . . . . . . . . . . . 131
An Introduction to Noncommutative Geometry
JohnMadore . . . . . . . . . . . . . . . . . . . . . . . . . 231
Geometric Properties of Transport in
Quantum Hall Systems
ThomasRichter and RuediSeiler . . . . . . . . . . . . . . . . 275
q-Deformed Heisenberg Algebras
JuliusWess . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Abstracts of the Seminars
(given by participants of the school) . . . . . . . . . . . . . . . 383
Notes on Equivariant Localization
AntonAlekseev
Institutionenf¨orTeoretiskFysik,UppsalaUniversitet,Box803,S-75108,Uppsala,
Sweden
Abstract. We review the localization formula due to Berline-Vergne and Atiyah-
Bott, with applications to the exact stationary phase phenomenon discovered by
Duistermaat-Heckman. We explain the Weil model of equivariant cohomology and
recall its relation to BRST. We show how to quantize the Weil model, and obtain
new localization formulas which, in particular, apply to Hamiltonian spaces with
group valued moment maps.
1 Introduction
The purpose of these lecture notes is to present the localization formulas
for equivariant cocycles. The localization phenomenon was (cid:12)rst discovered
by Duistermaat and Heckman in [DH], and then explained in the works of
Berline-Vergne[BV]andAtiyah-Bott[AB].Themainideaofthelocalization
formulas is similar to the residue formula: a multi-dimensional integral is
evaluated exactly by summing up a number of the (cid:12)xed point contributions.
In Section 2 we review the localization formula of [BV] and [AB]. We use
an elementary example of the sphere S2 as an illustration. Then, we outline
the relation between the localization formulas and Hamiltonian Mechanics,
and recover the Duistermaat-Heckman formula [DH].
In Section 3 we discuss the relations between the localization formulas
and the group actions. In the case of the Duistermaat-Heckman formula,
localization is intimately related to the symmetry group of the underlying
Hamiltoniansystem.Inparticular,wecomparetheequivariantdi(cid:11)erentialto
the BRST di(cid:11)erential.
InSection4weexplainhowtoquantizetheequivariantcohomology.This
Sectionisbasedonthepapers[AMM],[AM],[AMW1]and[AMW2].Weend
up by presenting the new localization formula which is derived in [AMW2].
Some simple applications of this new formula can be found in [P]. Section 4
is based on the joint works with A.Malkin, E.Meinrenken and C.Woodward.
These notes do not touch upon various applications of localization for-
mulas in Physics. Usually, one proceeds by extrapolating the localization
phenomenon to path integrals. Some of the most exciting examples of this
approach can be found in [MNP], [W2], [G], [BT], [MNS]. In fact, [W2] was
the original motivation for the formulas of Section 3.
I am grateful to the organizers and participants of the 38th Schladming
Winter School for the inspiring atmosphere!
H. Gausterer, H. Grosse, and L. Pittner (Eds.): Proceedings 1999, LNP 543, pp. 1−24, 2000.
(cid:211) Springer-Verlag Berlin Heidelberg 2000
2 AntonAlekseev
2 Localization Formulas
InthisSectionwereviewthelocalization formuladuetoBerline-Vergne[BV]
andAtiyah-Bott[AB].Itisthenusedtoderivetheexactstationaryphasefor-
mula due to Duistermaat and Heckman [DH]. The presentation is illustrated
at the elementary example of sphere S2.
2.1 Stationary Phase Method
In this section we recall the stationary phase method. It applies when one is
interested in the asymptotic behavior at large s of the integral
(cid:90)
1
I(s)= dxeisf(x)g(x): (1)
−1
Hereweassumethatfunctionsf(x)andg(x)arereal,andsu(cid:14)cientlysmooth.
Atlarges>0theleadingcontributionintotheintegral(1)isgivenbythe
neighborhoodofthecriticalpointsoff(x),whereitsderivativeinxvanishes.
Let x be such a critical point. Then, one can approximate f(x) near x by
0 0
the (cid:12)rst two terms of the Taylor series,
1
f(x)=f(x )+ f00(x )(x−x )2+:::;
0 2 0 0
where ::: stand for the higher order terms.
The leading contribution of the critical point x into the integral I(s) is
0
given by a simpler integral
(cid:90)
1
I0(s)=g(x0)eisf(x0) dxe2isf(cid:48)(cid:48)(x0)(x−x0)2:
−1
This integral is Gaussian, and can be computed explicitly,
(cid:18) (cid:19)
I0(s)=g(x0)ei(sf(x0)+επ4) sjf020((cid:25)x )j 12
0
Here " is the sign on the second derivative f00(x ).
0
A similar formula holds for multi-dimensional integrals,
(cid:90)
I(s)= dnxg(x)eisf(x): (2)
Again, the leading contribution into the asymptotics at large s is given by
thecriticalpointsoff(x),whereitsgradientvanishesrf =0.Atthecritical
point x one can expand f(x) into the Taylor series,
0
1 (cid:88)n
f(x)=f(x )+ f00(x )(x−x ) (x−x ) +:::; (3)
0 2 ij 0 0 i 0 j
i,j=1
Notes on Equivariant Localization 3
where
@2f
f00 = :
ij @x @x
i j
We assume that the critical point is non-degenerate, that is, the matrix f00
ij
is invertible. Then, the leading contribution of x into the integral I(s) is of
0
the form,
(cid:18) (cid:19)
I0(s)=g(x0)eisf(x0) 2s(cid:25) n2 jdet(fei0σ0(π4x ))j1: (4)
0 2
Here (cid:27) =(cid:27) −(cid:27) is the signature of the matrixf00, (cid:27) and (cid:27) are numbers
+ − ij + −
of positive and negative eigenvalues, respectively.
In general, one can have several critical points. Then, on can add the
leading contributions (4) to obtain the approximate answer for I(s),
(cid:18) (cid:19)
I(s)(cid:25) 2s(cid:25) n2 (cid:88)g(xi)eisf(xi) jdet(efiσ00i(πx4 ))j1 (5)
i i 2
Ofcourse,thereisnoreasonfortherighthandsidetobetheexactanswerfor
I(s).Butsometimesthisisthecase!Suchasituationiscalledexactstationary
phase, and will be studied in these notes.
Example: sphere S2 The simplest example of the exact stationary phase
phenomenon is the computation of the following integral. Consider the unit
sphere S2 de(cid:12)ned by equation x2+y2+z2 =1. We choose g(x;y;z)=1 and
f(x;y;z)=z. Then, the integral I(s) is of the form,
(cid:90)
I(s)= dA eisz; (6)
S2
(cid:82)
where dA is the area element normalized in the standard way, dA=4(cid:25).
S2
The critical points of the function f(x;y;z) = z are the North and the
South poles of the sphere. At both points one can use x and y as local
coordinates to obtain,
1
z (cid:25)1− (x2+y2)
2
near the North pole, and
1
z (cid:25)−1+ (x2+y2)
2
near the South pole. Thus, for the stationary phase approximation one ob-
tains,
2(cid:25) sin(s)
I(s)(cid:25) (−ieis+ie−is)=4(cid:25) : (7)
s s
HerewehaveusedthatatbothNorthandSouthpolesdet(f )=1,andthat
ij
(cid:27) =−2 and (cid:27) =2.
N S